reserve n,i,j,k for Nat;
reserve T for TuringStr,
  s for All-State of T;

theorem
  SumTuring computes [+]
proof
  now
    let s be All-State of SumTuring,t be Tape of SumTuring, h1 be Element of
    NAT, x be FinSequence of NAT;
    assume that
A1: x in dom [+] and
A2: s=[the InitS of SumTuring,h1,t] and
A3: t storeData <*h1*>^x;
    x is Tuple of 2,NAT by A1,Th28,FINSEQ_2:131;
    then consider i,j being Element of NAT such that
A4: x = <*i,j*> by FINSEQ_2:100;
A5: s = [0,h1,t] by A2,Def14;
A6: <*h1*>^x=<*h1,i,j*> by A4,FINSEQ_1:43;
    hence s is Accept-Halt by A3,A5,Th27;
    take h2=1+h1;
    take y=i+j;
    t storeData <*h1,i,j *> by A3,A4,FINSEQ_1:43;
    hence (Result s)`2_3=h2 by A5,Th27;
    thus y=[+].x by A4,COMPUT_1:85;
    (Result s)`3_3 storeData <*1+h1,i+j *> by A3,A5,A6,Th27;
    hence (Result s)`3_3 storeData <*h2*>^<* y *>;
  end;
  hence thesis;
end;
